We will examine the following research problems: 1. Model Identifiability and Posterior Propriety for Generalized Linear Models (GLMs) with Ignorably and Nonignorably Missing Covariates - We will carry out a theoretical investigation for establishing necessary and sufficient conditions for posterior propriety and existence of the maximum likelihood estimator for GLMs with ignorably nor nonignorably missing covariates. Our work will then be extended to other types of models including i) the Cox regression model and ii) generalized linear mixed models. 2. Model Assessment and Sensitivity Analyses in Missing Data Problems - We will derive model selection tools for assessing models and carrying out sensitivity analyses in missing data problems. The model assessment criteria will be quite general, and can be used to assess goodness of fit and sensitivity analyses in the presence of nonignorably missing covariate and/or response data in generalized linear models, models for longitudinal data, and survival models. 3. Theory and Inference for the Cox Regression Model with Missing Covariates - A theoretical investigation for inference with missing covariate data using Cox's partial likelihood will be conducted. Necessary and sufficient conditions for the existence of the maximum partial likelihood estimate (MPLE) will be studied for complete data settings as well as with MAR covariates. In addition, Bayesian methods for MAR covariate data will be studied. 4. Semiparametric and Nonparametric Specification of the Covariate Distribution and Missing Data Mechanism - We will examine semiparametric models for the covariate distribution and missing data mechanism for regression models with ignorably or nonignorably missing covariate and/or response data.